1. Confidence Intervals for the Regression Slope 12.1b Target Goal: I can perform a significance test about the slope β of a population (true) regression line. h.w: Pg 761: 13 - 19 odd

2. Review Linear Regression

3. Height Weight How much would an adult male weigh if he were 5 feet 4 inches tall? Weights of men will vary – in other words, there is a distribution of weights for adult males who are 5 feet 4 inches tall. This distribution is normally distributed. We want the standard deviations of all these normal distributions to be the same. Let’s look at the heights and weights of a population of adult men. Are some of these weights more likely than others? What would this distribution look like? What would you expect for other heights? Where would you expect the population regression line to be? 7072

4. The Sampling Distribution of b : Old Faithful cont. The Sampling Distribution of b : Old Faithful cont. Inference for Linear Regression Let’s return to our earlier exploration of Old Faithful eruptions. For all 222 eruptions in a single month, the population regression line for predicting the interval of time until the next eruption y from the duration of the previous eruption x is µ y = 33.97 + 10.36x. The standard deviation of responses about this line is given by σ = 6.159. If we take all possible SRSs of 20 eruptions from the population, we get the actual sampling distribution of b. Shape: Center : (b is an unbiased estimator of β) In practice, we don’t know σ for the population regression line. So we estimate it with the standard deviation of the residuals, s. Then we estimate the spread of the sampling distribution of b with the standard error of the slope: Approx. Normal µ b = β = 10.36

5. Performing a Significance Test for the Slope Performing a Significance Test for the Slope Inference for Linear Regression Suppose the conditions for inference are met. To test the hypothesis H 0 : β = hypothesized value, compute the test statistic Find the P-value by calculating the probability of getting a t statistic this large or larger in the direction specified by the alternative hypothesis H a. Use the t distribution with df = n - 2. Suppose the conditions for inference are met. To test the hypothesis H 0 : β = hypothesized value, compute the test statistic Find the P-value by calculating the probability of getting a t statistic this large or larger in the direction specified by the alternative hypothesis H a. Use the t distribution with df = n - 2. Typically = 0

6. Example: Crying and IQ Example: Crying and IQ Inference for Linear Regression Infants who cry easily may be more easily stimulated than others. This may be a sign of higher IQ. Child development researchers explored the relationship between the crying of infants 4 to 10 days old and their later IQ test scores. A snap of a rubber band on the sole of the foot caused the infants to cry. The researchers recorded the crying and measured its intensity by the number of peaks in the most active 20 seconds. They later measured the children’s IQ at age three years using the Stanford-Binet IQ test. A scatterplot and Minitab output for the data from a random sample of 38 infants is below. Do these data provide convincing evidence that there is a positive linear relationship between crying counts and IQ in the population of infants?

7. Example: Crying and IQ Example: Crying and IQ Inference for Linear Regression State: We want to perform a test of H 0 : β = 0 H a : β > 0 where β is the true slope of the population regression line relating crying count to IQ score. No significance level was given, so we’ll use α = 0.05. Do these data provide convincing evidence that there is a positive linear relationship between crying counts and IQ in the population of infants?

8. Example: Crying and IQ Example: Crying and IQ Inference for Linear Regression Plan: If the conditions are met, we will perform a t test for the slope β. Linear: The scatterplot suggests a moderately weak positive linear relationship between crying peaks and IQ. The residual plot shows a random scatter of points about the residual = 0 line. Independent: Later IQ scores of individual infants should be independent. Due to sampling without replacement, there have to be at least 10(38) = 380 infants in the population from which these children were selected.

9. Example: Crying and IQ Example: Crying and IQ Inference for Linear Regression Normal: The Normal probability plot of the residuals shows a slight curvature, which suggests that the responses may not be Normally distributed about the line at each x-value. With such a large sample size (n = 38), however, the t procedures are robust against departures from Normality. Equal variance: The residual plot shows a fairly equal amount of scatter around the horizontal line at 0 for all x-values. Random: We are told that these 38 infants were randomly selected.

10. Example: Crying and IQ Example: Crying and IQ Inference for Linear Regression Do: With no obvious violations of the conditions, we proceed to inference. The test statistic and P-value can be found in the Minitab output. Conclude: The P-value, 0.002, is less than our α = 0.05 significance level, so we have enough evidence to reject H 0 and conclude that there is a positive linear relationship between intensity of crying and IQ score in the population of infants. The Minitab output gives P = 0.004 as the P-value for a two-sided test. The P-value for the one-sided test is half of this, P = 0.002.

11. Testing the Hypothesis of No Linear Relationship To test whether or not there is a correlation between two quantitative variables, consider the slope of the regression line. To test whether or not there is a correlation between two quantitative variables, consider the slope of the regression line. If there is no correlation, the slope would be zero. If there is no correlation, the slope would be zero. H 0 : β = 0 (The mean of y does not change at all as x changes.)

12. To test this hypothesis, compute the t statistic and P-value. To test this hypothesis, compute the t statistic and P-value.Note: Regression output from statistical software usually gives t and its two- sided P-value. Regression output from statistical software usually gives t and its two- sided P-value. For a one-sided test, divide the P-value in the output by 2. For a one-sided test, divide the P-value in the output by 2.

13. Ex: Beer and Blood Alcohol How well does the number of beers a student drinks predict his or her blood alcohol content? How well does the number of beers a student drinks predict his or her blood alcohol content? Sixteen of age college student volunteers drank a randomly assigned number of cans of beer. Thirty minutes later, a police officer measured their blood alcohol content (BAC). Sixteen of age college student volunteers drank a randomly assigned number of cans of beer. Thirty minutes later, a police officer measured their blood alcohol content (BAC).

14. Data revealed: They noticed a variation in the data and didn’t believe that the number of drinks predicted the BAC well. Here is the scatter plot. They noticed a variation in the data and didn’t believe that the number of drinks predicted the BAC well. Here is the scatter plot. The solid line is the LSRL. The solid line is the LSRL. The scatterplot shows a clear linear relationship. The scatterplot shows a clear linear relationship.

15. Minitab output: Because r 2 = 0.8000, the number of drinks accounts for 80% of the observed variation in BAC. (The students are wrong).

16. To test hypothesis that the number of beers has no effect on BAC; H o : β = 0 There is no correlation between the number of beers consumed and BAC. H a : β > 0 There is a positive correlation between the number of beers consumed greater the BAC.  Examine the P-value  From the output: P-value = 0.0000,  The one sided P-value is half of this so it is also close to 0.

17. Thus, we reject H o and conclude that the number of beers does have an effect on BAC. Thus, we reject H o and conclude that the number of beers does have an effect on BAC. Calculator (page 804): LinRegTest Calculator (page 804): LinRegTest (Do for “crying vs. IQ.) (Do for “crying vs. IQ.)